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A Three-State Node Reliability Model for Sensor Networks M.H. Shazly, E.S. Elmallah

H.M.F. AboElFotoh

Department of Computing Science University of Alberta Edmonton, T6G 2H1, Canada E-Mail: {shazly,ehab}@cs.ualberta.ca

Department of Mathematics and Computer Science, Kuwait University P.O. Box 5969 Safat, Kuwait 13060 E-mail: [email protected]

Abstract—In this paper we formulate and analyze a model for assessing the reliability of a wireless sensor network (WSN) based on classifying the operating states of each node at any instant into one of three possible states: a state where both the sensing and wireless modules are operating, a state where only the wireless module is operating, and a state where the wireless module is failed. Thus, in the second state a node can only relay traffic among its neighbours without generating its own data. We define the reliability of a WSN as the probability that the sink node can collect data from a number of nodes whose total weight exceeds a specified threshold limit, given that each node can be in any one of the three possible states with a given probability. Existing results in the literature show that a restricted 2-state version of the problem is #P-hard even when the network is a rectangular grid. Nevertheless, for a rectangular W × L grid on n nodes where the sink node lies in one of the corners, the restricted 2-state reliability problem can be solved in O(nL2W ) time. Thus, the algorithm runs in polynomial time for any fixed W . Our work here derives an exact algorithm for the generalized 3-state reliability model on a generalized class of grids, called diagonalized grids, while maintaining the same O(nL2W ) running time. We obtain numerical results that illustrate the use of the devised algorithm as a WSN topological design tool.

I. I NTRODUCTION Wireless sensor networks (WSNs) are currently receiving increasing attention as a means for providing new solution platforms to various monitoring and control problems that arise in diverse application domains. Currently, there is a vast amount of literature investigating key areas required for successful design of dependable and energy-efficient sensor networks (see, e.g., the surveys in [1]–[5]). Quantifying dependability on WSNs, however, requires derivation of reliability figures. This is especially true since WSNs are typically conceived to operate in harsh environments. In such environment, the wireless module of a node may temporarily fail to work if it encounters a persisting channel fading condition that disrupts communication. Similarly, the sensing module may temporarily fail to provide useful data if a node’s sensing field is obstructed. In addition, sensor nodes are expected to have low manufacturing costs and operate on limited energy supplies. Thus, component failure or energy depletion may cause the wireless module or the sensing module to fail for prolonged amount of time. The work presented in [6]–[9] focuses on WSN reliability analysis.

In contrast with previous work where a sensor node can either operate or fail, our work here adopts a 3-state node reliability model where the effect of wireless module failure is distinguished from the effect of sensing module failure. Our objective is to develop more accurate network reliability models. Using the adopted 3-state node model, we formalize a problem, called WSN3REL, where sensor nodes are assigned different weights to model the relative importance of getting data from different sensor nodes. After an event of random failure, a network is considered operational if the sink node can collect data from a subset of nodes whose total weight exceeds a specified threshold value. In [8], the authors define a 2-state reliability problem called WSNREL that is a special case of the WSN3REL problem. WSNREL has been shown in [8] to be #P-hard. Consequently, our present WSN3REL problem is also #Phard. Such intractability results motivate searching for special classes of networks for which the problem can be solved efficiently, as well as methods for computing effective upper and lower bounds on the exact solution. In [8], the authors present an exact algorithm for solving the WSNREL problem on the well know class of sensor grids [10]–[14] where the sink node lies in one corner. The algorithm runs in time O(nL2W ) on a W × L grid where n is the number of nodes. Our main contribution in this paper is an exact algorithm for solving the more general WSN3REL on a class of WSNs, called diagonalized grids (or d-grids for short), that includes sensor grids as a subclass. The algorithm presented here has the same time complexity as that in [8]. II. S YSTEM M ODEL AND P ROBLEM F ORMULATION A. Three-State Node Reliability Model We consider WSNs with a single sink node. We model the flow of traffic from sensor nodes to the sink node by a probabilistic directed graph G = (V, E) where V is the set of sensor nodes, and E is a set of directed links. For a sensor node v, we denote by pcom (v) the probability that the wireless module operates. Similarly, we denote by psense (v) the probability that the sensing module operates. We assume that the wireless module fails independently from the sensing module. (Energy depletion is assumed to fail the wireless module long before affecting the sensing module.) Thus a node

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v can be in one of three possible states with the following probabilities: pcs (v) = pcom (v) × psense (v): the probability the node can perform both communication and sensing, pc (v) = pcom (v) × (1 − psense (v)): the probability that the node can communicate but not sense the environment, and q(v) = 1 − pcs (v) − pc (v): the probability that the node cannot communicate with any other node. In addition, we assume that the WSN uses an adaptive routing algorithm that can utilize any existing operational directed path from any node v to the sink node. B. The Reliability Function We associate with each node v a weight wv that represents the relative importance of the data samples produced by node v. Thus, e.g., data generated from certain geographic areas covered by the sensor network can be assigned higher importance level than data generated from other areas. A network is operating if the sum of the weights of nodes whose generated traffic reach the sink exceeds a user specified threshold value, denoted Wreq . Otherwise, if the sum of such weights falls below Wreq , the sampling of the data are deemed to be insufficient to monitor the phenomenon of interest, and the network is considered failed. Our problem can then be defined as follows. Definition (the WSN3REL problem): Given a WSN graph G = (V, E) with a distinguished sink node, a threshold value Wreq , and for each non-sink node v, a weight wv , and the operating probabilities pc (v) and pcs (v), compute the probability Rel(G) that there exists a subset of operational nodes S ⊆ V whose own generated traffic reach the sink and  v∈S wv ≥ Wreq . C. Complete State Enumeration A complete state enumeration algorithm for computing Rel(G) of a network G on n nodes can be formalized as follows. Following an event of random failure or severe degradation in the sensing or communication capabilities of some nodes, the network can be in any one of O(3n ) states. Each state T = (Tcs , Tc ) is defined by a subset Tcs of nodes that can perform both sensing and relaying, and another relaying only. disjoint subset Tc of nodes that can perform  Each of the remaining nodes in T \ {Tcs Tc } is in a failed state where the node cannot communicate with any of its neighbours. wa=4 a

wb=3 b

TABLE I: Sample operating states State T1 = ({a, c}, {d}) T2 = ({a, b, c}, φ) T3 = ({a, c, e}, φ)

State Probability and Weight Prob [T1 ] = pcs (a)pcs (c) pc (d) q(b)q(e) W (T1 ) = wa + wc = 5 Prob [T2 ] = pcs (a)pcs (b)pcs (c) q(d)q(e) W (T2 ) = wa + wb + wc = 8 Prob [T3 ] = pcs (a)pcs (c)pcs (e) q(b)q(d) W (T3 ) = wa = 4

probabilities and objective function values. Note that in state T3 nodes b and d form a cutset that forbids traffic from nodes c and e to reach the sink.  By definition, state T is operational if thesubset Tcs ⊆ Tcs of nodes that can reach the sink satisfies: v∈Tcs  wv ≥ Wreq . Denote by    pcs (a) pc (a) q(a) Prob [T ] =  a∈T a∈T cs

a∈{T / cs

c

the probability that state  T arises. Thus, the required probability Rel(G, Wreq ) = Prob [T ], where T is an operational state with respect to the specified aggregate value Wreq . Such complete state enumeration algorithm requires O(3n ) time to compute Rel(G). Hence, it can be used only for networks of limited size. D. Diagonalized Grids (D-Grids) The main algorithm presented in the next section computes Rel(G) for a class of single sink sensor networks called diagonalized grids (or d-grids for short). In a W × L dgrid network G, each node is assigned non-negative integer (x, y) coordinates, where x ∈ [0, L − 1] and y ∈ [0, W − 1], and the sink node is placed at the origin (0, 0). Thus, G has n = W × L nodes. In addition, G has diagonal links of the form ((x, y), (x − 1, y − 1)) where x, y ≥ 1. Thus, a W × L d-grid network, W, L ≥ 2, has (W − 1) × (L − 1) more links than a complete W × L grid network. We note that compared to a grid, (a) utilizing diagonal links in a d-grid reduces the number of hops to the sink while requiring a relatively small increase in a node’s transmission power, (b) a d-grid has more routes from any node to the sink, and (c) a d-grid provides a better means of approximating the structure of an arbitrary network of the same size. Thus, d-grids are expected to be more power efficient and reliable than grids. The following example illustrates the use of d-grids in approximating the structure of arbitrary single sink WSNs.

wc=1 c

(1,3)

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(0,3)

(3,2) (1,2)

sink node

e we=3

(5,2)

(2,2) (4,1)

(0,2)

d wd=4

Ts }

(2,1) (1,1)

Fig. 1: An example WSN Example. Fig. 1 illustrates a network on 6 nodes where each node has the indicated weight, and traffic to the sink can utilize any of the directed links shown. The network has 35 different states. Table I shows three possible states along with their

(-3,0)

(-3,-1)

(-2,0)

(0,1) (-1,0) (-1,-1)

sink (0,0)

(4,0) (2,0)

(5,0)

(3,0)

(1,0) (0,-1)

(-2,-1)

(5,1)

(3,1)

(0,-2)

(-1,-2)

Fig. 2: Decomposition of a WSN into two d-grids Example. Fig. 2 illustrates an arbitrary WSN with a single sink

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node. Nodes that are assigned labels {(i, j) : i ≥ 0, j ≥ 0)} form an incomplete 4 × 6 d-grid G1 where diagonal links (of the form ((i, j), (i − 1, j − 1))) appear as dashed lines. Compared to a complete d-grid, G1 is missing some links, e.g., links ((1, 3), (1, 2)), ((3, 2), (3, 1)). Similarly, nodes that are assigned labels {(i, j) : i ≤ 0, j ≤ 0)} form another incomplete diagonalized grid G2 . We remark that decomposing an arbitrary network into a few incomplete d-grids (as done in the above example) can be used in conjunction with the algorithm presented in the next section to derive a lower bound on Rel(G). III. T HE M AIN A LGORITHM Given a W × L d-grid network G = (V, E), a 3-state probability model for each node, and a required weight Wreq , we present an exact algorithm for computing Rel(G) (function DG3REL in Fig. 4). Each node (x, y) in the d-grid has x ∈ [0, L−1] and y ∈ [0, W −1]. So, G has n = W ×L nodes. We assume, without loss of generality, that W ≤ L and the sink node is at the origin (0, 0). The algorithm then processes the nodes in an increasing order of their X-coordinates, and for nodes that have the same X-coordinate, in an increasing order of their Y -coordinates. Thus, nodes are processed according to a total ordering defined by the relation: (xi , yi ) < (xj , yj ) if xi < xj , or when xi = xj but yi < yj . A typical iteration of the algorithm processes node (x, y) by generating and consolidating information about the states of the subgrid formed by the nodes V≤(x,y) = {(0, 0), (0, 1), . . . , (x, y)}. We denote such subgrid by G≤(x,y) . The algorithm strives to achieve efficient execution time by avoiding generation of a number of states that grows exponentially with n. To this end, the function consolidates information about many network states that are considered of the same type. To explain this consolidation process further, we introduce the following definitions. Definition (boundary nodes of a subgrid): For a given node (x, y), the boundary of subgrid G≤(x,y) is the subset of nodes that can be reached by nodes outside the subgrid (i.e., nodes in G \ G≤(x,y) ) using exactly one link. Example. Fig. 3 illustrates a d-grid of width W = 6, and length L = 6. Assume that the algorithm is at the iteration where node (x = 2, y = 3) is considered. The associated subgrid G≤(x,y) has 16 nodes; the remaining nodes in V \ V≤(x,y) , shown as dotted circles, are still to be processed in subsequent iterations. Using x = 2 and y = 3, the boundary of G≤(x,y) is the set of nodes {(x, 0), (x, 1), (x, 2), (x, 3), (x − 1, 3), (x − 1, 4), (x − 1, 5)} shown inside the rectangles. Definition (state type): Let T be a state (i.e., a set of operating nodes) in a subgrid G≤(x,y) . The type of T in G≤(x,y) is a triple (α, β, f ) where - α is the set of Y -coordinates of boundary nodes of G≤(x,y) that lie in column x − 1 and can reach the sink by the specified routing algorithm, - β is the set of Y -coordinates of boundary nodes that lie in column x (if such column exists) and can reach the

sink by the specified routing algorithm, and - f is the sum of node weights whose own generated traffic reaches the sink when G≤(x,y) is in state T . boundary nodes Y α = {3,5} 5 X 4

X

β ={1,2} (x=2,y=3)

3

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2 1 0

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sink node fully operating node X failed node relay node not yet considered

Fig. 3: A subgrid G≤(x,y) and its boundary nodes Example. In Fig. 3 assume that the algorithm is processing node (x = 2, y = 3). The set of 11 non-sink operating nodes in G≤(x,y) forms a state T of type (α, β, f ) where α = {3, 5} since relay nodes (x−1, 3) and (x−1, 5) lie on the boundary of G≤(x,y) and can reach the sink, and β = {1, 2} since nodes (x, 1) and (x, 2) are operating and can reach the sink. In contrast, boundary nodes (x, 0) is operational, but cannot reach the sink, and node (x, 3) is failed. Hence, by our definition, set β does not contain {0, 3}. Moreover, in state T only 3 nodes generate data that can reach the sink. Assuming equal node weights, then f = 3. Thus, T is of type (α = {3, 5}, β = {1, 2}, f = 3). A. Algorithm Description To explain function DG3REL, we introduce the following notation. Let (v0 , v1 , . . . , vn−1 ) be the listing of the nodes according to the total order defined above where node v0 is the sink node, and for any index i, node vi = (xi , yi ). For simplicity, we denote subgrid G≤(xi ,yi ) by Gi . In addition, for each node v = (x, y), we denote by down (x, y), diag (x, y), and lef t (x, y) the possible directed links connecting node (x, y) to its neighbours at coordinates (x, y −1), (x−1, y −1), and (x − 1, y), respectively. Some of these links may not exist if (x, y) is a boundary node (where either x = 0, or y = 0), or if G is a partial d-grid. The function uses two arrays R and Q to store mappings form state types to probability values, as explained next. In step 1, the function processes the sink node v0 = (0, 0) by initializing R(α = φ, β = {0}, f = 0) = 1. The state type (α = φ, β = {0}, f = 0) corresponds to the only network state that arises in subgrid G0 . This follows since the sink node is always operational (hence, the Y -index 0 is added to β), but the sink does not generate its own data (hence f = 0).

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Function DG3REL(G, W, L, pcs , pc , Wreq ): Input: a possibly incomplete W ×L d-grid G where each sensor node (x, y) has parameters wx,y , pcs (x, y), and pc (x, y). Output: Rel, the probability that the total weight of nodes whose generated traffic reach the sink is at least Wreq . Notation: q(x, y) = 1 − pcs (x, y) − pc (x, y) Data Structures: arrays R and Q 1. initialize R with one state type for the sink node: R(α = φ, β = {0}, f = 0) = 1 2. for each node (x, y) in the total order described above { 3. if ((x, y) == the sink node) continue 4. if (y == 0) rename each state (α, β, f ) in R to be (β, φ, f ) 5. for each state (α, β, f ) in R { 6. Q(α, β, f ) + =R(α, β, f ) ∗ q(x, y) 7. if ( (y ∈ α AND llef t (x, y) exists) OR (y − 1 ∈ α AND ldiag (x, y) exists) OR (y − 1 ∈ β AND ldown (x, y) exists) ) { 8. Q(α, β + y, min(f + wx,y , Wreq )) + = R(α, β, f ) ∗ pcs (x, y) 9. Q(α, β + y, f ) + = R(α, β, f ) ∗ pc (x, y) 10. } 11. else Q(α, β, f ) + = R(α, β, f ) ∗ (1 − q(x, y)) 12. } 13. for each state (α, β, f ) in Q remove element y−1 from α (if it exists) 14. set R = Q; clear Q 15. }  16. return Rel = R(α, β, f ) f ≥Wreq

and relaying and whose own traffic can reach the sink to state T results in state (α, β + y, min(f + wx,y , Wreq )). 3) Step 9: adding vi as a relay node that can reach the sink to T results in state (α, β + y, f ). 4) Step 11: adding vi as an operating node that cannot reach the sink results in state (α, β, f ). After finishing the loop in Step 5, in Step 13 we omit index y − 1 from each set α since node (x − 1, y − 1) is not a boundary node of subgrid Gi+1 . Step 14 stores the computed probabilities in R, and clears Q for the next iteration. B. Running Time The maximum size of array R (or Q) is Wreq × 2W +1 . Thus, a rough upper bound on the running time of the function is O(nWreq 2W +1 ). We remark that when all nodes have the same weight, after processing node (x, y), an upper bound on the number of distinct f values in array R (or Q) is W (x+1). Hence, an upper bound on the number of state types in array bound on the number of R is 2W +1 W (x + 1). Thus, an upper x=L−1 steps executed by the algorithm is x=0 2W +1 W (x+1) = O(nL2W ), as achieved by the algorithm in [8]. IV. N UMERICAL R ESULTS In this section we illustrate the usefulness of our 3-state reliability evaluation algorithm by examining the following basic aspects: (a) the improved accuracy gained by utilizing the 3-state model, (b) the improved reliability bound gained by using d-grids, (c) the impact of improving the reliability of the sensing modules on Rel(G), and (d) the impact of improving the reliability of the wireless modules on Rel(G).

The ith, i ≥ 1, iteration of the for-loop in step 2 processes node vi = (xi , yi ). At the start of the iteration, each entry R(α, β, f ) in array R contains the probability that subgrid Gi−1 is in a state of type (α, β, f ). That is,

Reliability

Fig. 4: Pseudo-code for function DG3REL

R(α, β, f ) = Prob [T : T is a state of Gi−1 of type (α, β, f )] .

2−state model 3−state model Pcom = 0.9 Psense= 0.8

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Fig. 5: Improvement gained by using the 3-state model (200 sensors in a 10 × 20 grid)

Reliability

Step 4 renames each state (α, β, f ) in R to (β, φ, f ) prior to the processing of the bottom most node (x, y = 0) in a new column x. The renaming is required since the indices stored in a set α during the processing of the previous column (column x − 1) belong to column x − 2 which is no longer a column that contains boundary nodes when nodes in the current column x are considered. Subsequently, the inner forloop of Step 5 stores in array Q similar probabilities for Gi . An important ingredient in proving correctness is to show that Step 5 exhaustively generates all possible state types of Gi that contribute to computing Rel(G), and for each such state type, the function computes the corresponding aggregate probability correctly. To see this, we note that if T is a state of Gi−1 of type (α, β, f ) then 1) Step 6: adding vi as a failed node to T results in a state of the same type. 2) Step 8: adding vi as a node that can perform both sensing

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

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Fig. 6: Improvement gained by using d-grids over grids (25 sensors in a 5 × 5 field)

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0.6 D−Grid Reliability

utilize wireless modules of various grades of reliability.

Psense= 0.5 Psense= 0.6 Psense= 0.7 Psense= 0.8 Psense= 0.9 Psense= 1.0

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V. C ONCLUSION AND F UTURE W ORK

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Fig. 7: Effect of increasing the weight Wreq on Rel(G) for different sensing module operation probabilities (200 sensors in a 10 × 20 d-grid) D−Grid Reliability

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Pcom= 0.5 Pcom= 0.6 Pcom= 0.7 Pcom= 0.8 Pcom= 0.9 Pcom= 1.0

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Fig. 8: Effect of increasing the weight Wreq on Rel(G) for different wireless module operation probabilities (200 sensors in a 10 × 20 d-grid) Fig. 5 illustrates aspect (a) above. Here, we use a 10 × 20 grid where each node x has pcom (x) = 0.9, psense (x) = 0.8, and all nodes have equal weight. Thus, in a 2-state model, node x operates with probability pcom (x) × psense (x) = 0.72. The figure illustrates that for a wide range of Wreq (e.g., the range [20−140]) the 2-state model significantly underestimates Rel(G) by undermining the strong effect of using reliable wireless modules. Fig. 6 illustrates aspect (b) above. Here, we consider a nongrid network G where each node x has pcom (x) = 0.6, and psense (x) = 0.9. G has a 5 × 5 d-grid subgraph G1 , and a 5 × 5 grid subgraph G2 . Thus, Rel(G1 ) and Rel(G2 ) can be used as lower bounds on Rel(G). Fig. 6, however, illustrates that for Wreq = 5, G1 improves over G2 by more than 30% in estimating Rel(G) (from 0.6 to 0.8). Thus, bounds obtained using approximations to d-grids can be significantly better than bounds obtained using approximations to grids. We remark (from additional results not shown in the figure) that the gap between Rel(G1 ) and Rel(G2 ) decreases as pcom increases. Fig. 7 illustrates aspect (c) above. Here, for each node x in a 10 × 20 d-grid, pcom (x) = 0.5, while psense (x) varies in the range [0.5 − 1.0]. The curves quantify the intuition that the cost incurred in improving the reliability of the sensing modules may not be justified if the wireless modules are of low quality. In contrast, Fig. 8 illustrates aspect (d) above, where for each node x, psense (x) = 0.5 while pcom (x) varies in the range [0.5 − 1.0]. Again, the curves quantify the intuition that improving the quality of the wireless modules are of prime importance. We recall that our algorithm allows each node v to have a specified probability pc (v). Thus, our algorithm provides a tool for analyzing networks where sensor nodes

With the growing use of WSNs in different sensitive applications there is a growing need to analyze the reliability of such networks. In this paper, we adopt a 3-state reliability model for sensor nodes where the role played by fully operational nodes is distinguished from the role played by nodes that can only relay data. Our model allows each node to have a weight that represents the importance of the data generated by the node. We formalize a reliability measure that captures the network’s ability to aggregate data from sensor nodes whose total weight exceeds a specified threshold weight Wreq . We show an exact algorithm for computing the reliability of partial rectangular W ×L diagonalized grid networks. The worst case running time of the algorithm indicates that it is particularly efficient for thin grids of small width W . For future work, we propose investigating the use of the devised algorithm in bounding the reliability of more complex networks. Acknowledgment. This research is supported by NSERC Canada. We thank the reviewers for their helpful comments. R EFERENCES [1] R. Verdone, D. Dardari, G. Mazzini, and A. Conti, Wireless Sensor and Actuator Networks: Technologies, Analysis and Design. Academic Press, 2008. [2] A. A. Abbasi and M. Younis, “A survey on clustering algorithms for wireless sensor networks,” Computer Communications, vol. 30, no. 1415, pp. 2826–2841, 2007. [3] L. Wang and Y. Xiao, “A survey of energy-efficient scheduling mechanisms in sensor networks,” Mobile Networks and Applications, vol. 11, no. 5, pp. 723–740, 2006. [4] D. J. Vergados, N. A. Pantazis, and D. D. Vergados, “Energy-efficient route selection strategies for wireless sensor networks,” Mobile Networks and Applications, vol. 13, no. 3-4, pp. 285–296, 2008. [5] P. Santi, “Topology control in wireless ad hoc and sensor networks,” ACM Computing Surveys, vol. 37, no. 2, pp. 164–194, 2005. [6] H. AboElFotoh, S. Iyengar, and K. Chakrabarty, “Computing reliability and message delay for cooperative wireless distributed sensor networks subject to random failures,” IEEE Transactions on Reliability, vol. 54, pp. 145–155, March 2005. [7] H. AboElFotoh, E. Elmallah, and H. Hassanein, “A flow-based reliability measure for wireless sensor networks,” International Journal of Sensor Networks, vol. 2, no. 5/6, pp. 311–320, 2007. [8] H. AboElFotoh and E. Elmallah, “Reliability of wireless sensor grids,” in Proceedings of the 33rd IEEE Conference on Local Computer Networks (LCN 2008), Montreal, 2008. [9] A. Shrestha, L. Xing, and H. Liu, “Modeling and evaluating the reliability of wireless sensor networks,” in Annual Reliability and Maintainability Symposium (RAMS), pp. 186–191, January 2007. [10] G. Barrenechea, B. Beferull-Lozano, and M. Vetterli, “Lattice sensor networks: capacity limits, optimal routing and robustness to failures,” in IPSN ’04, Berkeley, California, USA, ACM, 2004. [11] H. B. Lim, Y. M. Teo, P. Mukherjee, V. T. Lam, W. F. Wong, and S. See, “Integration of wireless sensor networks and the grid,” in Proceedings of the IEEE Conference on Local Computer Networks (LCN), pp. 91–99, November 2005. [12] K. Xu, G. Takahara, and H. Hassanein, “On the robustness of grid-based deployment in wireless sensor networks,” in IWCMC ’06, Vancouver, Canada, ACM, 2006. [13] Y. Zheng, D. J. Brady, and P. K. Agarwal, “Localization using boundary sensors: An analysis based on graph theory,” ACM Transactions on Sensor Networks (TOSN), vol. 3, no. 4, p. 21, 2007. [14] S. Shakkottai, R. Srikant, and N. Shroff, “Unreliable sensor grids: coverage, connectivity and diameter,” in INFOCOM 2003, March 2003.

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